Solving Trigonometric Equations: An Exploration of sin A 2sin 30°cos 30°

Trigonometry is a fundamental branch of mathematics that deals with the relationships between the angles and sides of triangles. Solving trigonometric equations, such as sin A 2sin 30°cos 30°, can provide valuable insights into trigonometric identities and properties. In this article, we will explore how to solve this equation step-by-step and find the possible values of angle A.

Step-by-Step Solution

To solve the equation sin A 2 sin 30° cos 30°, we can follow these steps:

First, let's calculate the values of sin 30° and cos 30°. sin 30° 1/2 cos 30° √3/2 Now, substitute these values into the equation.

sin A 2 (1/2) ( √3/2)

Let's simplify the expression. Therefore, the simplified equation is:

sin A 2 × 1/2 × √3/2 √3/2

Using Trigonometric Identities

The resulting equation, sin A √3/2, can be compared to the standard values of sine for commonly known angles. We know that:

sin 60° √3/2 sin 120° √3/2 sin (360° 60°) sin 420° √3/2 sin (360° 120°) sin 480° √3/2

Hence, the solutions for sin A √3/2 are:

A 60° 360°k A 120° 360°k

Exploring the Principal Values

For k 0, the principal values of A are:

A 60° A 120°

These solutions can be found in standard trigonometric tables or by using the double angle formula:

sin 2x 2sin x cos x

Applying the double angle formula to the given equation:

sin 2(30°) 2sin 30° cos 30°

Therefore:

sin 60° 2sin 30° cos 30° A 60°

Conclusion

In conclusion, the value of angle A, in the equation sin A 2sin 30° cos 30°, is:

A 60°, 120°, 420°, 480°, etc.

Additional Resources

For further reading and practice in trigonometry, consider consulting the following resources:

Trigonometry Textbooks for grades 9 and 10 Standard trigonometric tables Online trigonometry courses

By mastering these concepts, you can enhance your problem-solving skills and gain a deeper understanding of trigonometric equations and identities.